Behavioral scientists have shown that human beings hate a given percentage loss more than twice as much as they enjoy the same percentage gain. In a more general sense we quickly get used to higher levels of wealth and feel pain when our wealth declines from its highest level achieved. In the real world, then, it is drawdowns of wealth that matter. This is why downside volatility in a portfolio is so much more important than upside volatility.
For any portfolio (it could be a passive long-only stock portfolio or a hedge fund strategy) we can look at the historical growth of that portfolio, preferably on a daily basis over a long time period. On any day of that history we can look at the drawdown from the prior peak wealth. For example, on day 200 the portfolio might have a value 5% below the highest prior level. On day 201, after a gain of about 0.2%, the drawdown is now 4.8%, and so on. For every day there is an associated drawdown number, zero if the portfolio value just reached a new high. As an investor we would have felt the pain on each of these days, saying “I’m still down 4.8% from the peak.”
The Ulcer Index takes into account every one of those drawdown numbers. Specifically it is calculated as the square root of the average squared drawdown number. On day 200, the squared drawdown number is 5%*5%, on day 201 it is 4.8%*4.8% and so on. We add up all of these squared numbers, take the average of the result and then take the square root.
As can be seen by looking at the calculation, the Ulcer Index incorporates every single drawdown on every day of the history. A shallow drawdown that lasts a very long time will contribute a lot to the Ulcer Index because there will be a drawdown number for each of those days; a large drawdown will also be penalized a lot because of the squared factor (for example, a 10% drawdown has a weight of 10%*10% or 1% in the average versus a 5% drawdown at 0.25%).
The final number is a measure of the pain that was felt in the strategy over the period because it reflects all of the drawdown experience - not just the maximum drawdown but the frequency, magnitude and duration of drawdowns.
With the Ulcer Index, strategies which have lots of small gains but occasional large losses get penalized because the drawdowns are large relative to the ability to recover from those drawdowns, i.e., the drawdowns will tend to last a long time. In effect, the ulcer index takes into account the sequence of losses which is so important in the creation of drawdowns. Option selling strategies (including covered call methods for example) have this characteristic. Most of the time they work quite well and appear to have low volatility but then they have sudden sharp losses and it can take months or years of small gains to recover those losses.
Conversely, strategies which have more normal volatility or which are positively skewed (occasional large gains which quickly offset prior small losses) will have a lower Ulcer Index for a given return profile. These strategies have upside volatility, which is a good thing, and lower downside volatility, which is a very good thing.
Exhibit 1 illustrates the relationship between return and Ulcer Index for CTAs (primarily systematic trend-followers) as represented by the BTOP50 Index (1987-2009), the CS/Tremont Hedge Fund Index (1996-2009), the Endowment model (a proxy well-diversified portfolio similar to that held by endowments, 1987-2009) and a traditional 60/40 stock/bond portfolio (1970-2009). Clearly, the CTA index dominates the others from a mean/Ulcer Index (MU) point of view: higher return and lower risk. In effect the MU return per unit of ulcer pain is 2.7 for CTAs versus 1.9 for the hedge fund index, 1.1 for the endowment model and only 0.9:1 for the stock/bond portfolio.
Traditional mean/variance (MV) analysis often hides the reality of the sequence of returns and losses and their impact on investors. Variance weights upside and downside equally whereas the Ulcer Index focuses squarely on the drawdown experience, the experience we feel in our gut as our wealth erodes. We believe MU analysis is a more effective way of seeking an efficient portfolio than the old paradigm of MV.
From an investment management viewpoint, there tends to be an inverse relationship between the level of risk management in a strategy and the associated Ulcer Index. If a strategy rigorously controls the potential magnitude of bad outcomes (via stop-losses or other direct hedging mechanisms), the Ulcer Index tends to be low because it is hard for small drawdowns to become large drawdowns. Passive strategies such as buy and hold or short volatility do not control negative outcomes and hence can have arbitrarily large and lengthy drawdowns. These strategies have high realized Ulcer Indices over the long run and insufficient payoff for that pain.
The message to investors is clear. If you want to control your financial destiny and achieve returns with the least amount of pain, focus on the Ulcer Index of your strategies. Make sure you are getting an adequate return for the potential pain. Do not be passive about the magnitude of bad outcomes
Peter Matthews is the founder/managing partner of PJM Capital. Read our Trader Profile on Matthews here.